论文标题
异性高斯序列模型中的稀疏信号检测：尖锐的最小速率
Sparse Signal Detection in Heteroscedastic Gaussian Sequence Models: Sharp Minimax Rates
论文作者
论文摘要
给定具有未知平均值$θ\ in \ mathbb r^d $和已知的协方差矩阵$σ= \ operatorAtorName {diag}（σ_1^2，\ dots，σ_d^2）$的异质性高斯序列模型，我们研究了针对Sparse Sparsity $ S $ S $。也就是说，我们表征了应有多大的$ε^*> 0 $的特征，以便以较高的概率区分null假设$θ= 0 $与$ s $ s $ -s-sparse vectors组成的$ \ mathbb r^d $组成的替代品，与$ 0 $ 0 $ in $ n $ n in $ n $ n $ n $ n $ n n $ n n $ n n $ n $ n n $ nord（$ t $ t in [1 $ t \ in [1，\ infty $] $）。我们发现minimax上和下限在minimax分离半径$ε^*$上，并证明它们始终是匹配的。我们还得出相应的最小值测试，以达到这些界限。我们的结果揭示了有关$ε^*$在稀疏度，$ l^t $ metric和$σ$的异质性概况方面的$ε^*$行为的新相关。对于欧几里得人（即$ l^2 $）的分离，我们弥合了文献中其余的空白。
Given a heterogeneous Gaussian sequence model with unknown mean $θ\in \mathbb R^d$ and known covariance matrix $Σ= \operatorname{diag}(σ_1^2,\dots, σ_d^2)$, we study the signal detection problem against sparse alternatives, for known sparsity $s$. Namely, we characterize how large $ε^*>0$ should be, in order to distinguish with high probability the null hypothesis $θ=0$ from the alternative composed of $s$-sparse vectors in $\mathbb R^d$, separated from $0$ in $L^t$ norm ($t \in [1,\infty]$) by at least $ε^*$. We find minimax upper and lower bounds over the minimax separation radius $ε^*$ and prove that they are always matching. We also derive the corresponding minimax tests achieving these bounds. Our results reveal new phase transitions regarding the behavior of $ε^*$ with respect to the level of sparsity, to the $L^t$ metric, and to the heteroscedasticity profile of $Σ$. In the case of the Euclidean (i.e. $L^2$) separation, we bridge the remaining gaps in the literature.
